THE MIDPOINT AND DISTANCE FORMULAS

Midpoint Formula

Let A(x₁, y₁) and B(x₂, y₂).

The midpoint M of the line segment AB is

Finding the Coordinates of a Midpoint

Example 1 :

Find the coordinates of the midpoint of the line segment CD with endpoints C(-2, -1) and D (4, 2).

Solution :

Write the formula.

Substitute (-2, -1) for (x₁, y₁) and (4, 2) for (x₂, y₂). 

=  M[⁽⁻² ⁺ ⁴⁾⁄₂, ⁽⁻¹ ⁺ ²⁾⁄₂]

=  M(²⁄₂, ½)

=  M(1, ½)

Finding the Coordinates of an Endpoint

Example 2 :

M is the midpoint of the line segment AB. A has coordinates (2, 2), and M has coordinates (4, -3). Find the coordinates of B.

Solution :

Step 1 :

Let the coordinates of B equal (x, y).

Step 2 :

Use the Midpoint Formula.

(4, -3)  =  [⁽² ⁺ ˣ⁾⁄₂, ⁽² ⁺ ʸ⁾⁄₂]

Step 2 :

The coordinates of B are (6, –8).

Check : 

Graph points A and B and midpoint M.

Point M appears to be the midpoint of the line segment AB.

Pythagorean Theorem : 

c²  =  a² + b²

Solve for c. Use the positive square root to represent distance.

c  =  √(a² + b²)

This equation represents the Distance Formula.

Distance Formula

In a coordinate plane, formula to find the distance between two points (x₁, y₁) and (x₂, y₂) is

Finding Distance in the Coordinate Plane

Example 3 :

Use the Distance Formula to find the distance, to the nearest hundredth, from A(-2, 3) to B(2, -2).

Solution :

Distance Formula : 

d  =  √[(x₂ - x₁)² + (y₂ - y₁)²]

Substitute (-2, 3) for (x₁, y₁) and (2, -2) for (x₂, y₂). 

d  =  √[(2 + 2)² + (-2 - 3)²]

Simplify. 

d  =  √[4² + (-5)²]

d  =  √[16 + 25]

d  =  √41

d  ≈  6.40

The distance between from A(-2, 3) to B(2, -2) is about 6.40 units. 

Geography Application

Example 4 :

Each unit on the map of Lake Okeechobee represents 1 mile. Kemka and her father plan to travel from point A near the town of Okeechobee to point B at Pahokee. To the nearest tenth of a mile, how far do Kemka and her father plan to travel?

Solution :

Distance Formula : 

d  =  √[(x₂ - x₁)² + (y₂ - y₁)²]

Substitute (33, 13) for (x₁, y₁) and (22, 39) for (x₂, y₂). 

d  =  √[(33 - 22)² + (13 - 39)²]

Simplify. 

d  =  √[11² + (-26)²]

d  =  √[121 + 676]

d  =  √797

d  ≈  28.2

Kemka and her father plan to travel about 28.2 miles.

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